U.S. patent number 4,786,874 [Application Number 06/898,409] was granted by the patent office on 1988-11-22 for resistivity sensor for generating asymmetrical current field and method of using the same.
This patent grant is currently assigned to Teleco Oilfield Services Inc.. Invention is credited to Allen Duckworth, Donald S. Grosso.
United States Patent |
4,786,874 |
Grosso , et al. |
November 22, 1988 |
Resistivity sensor for generating asymmetrical current field and
method of using the same
Abstract
A device for detecting and sensing the boundaries between
differing rock strata is provided. This device utilizes electrical
conductivity or resistivity to detect and measure the boundary
layer. The device of the present invention may be used with a
"measurement-while-drilling" (MWD) tool, using either wire-line or
wireless data communication with a surface receiver. Moreover, the
resistivity or conductivity detection device of the present
invention may also be used alone or in conjunction with gamma
radiation sensors. The present invention utilizes at least a pair
of electrodes which occupy a small area and are located on one side
of the tool aligned with the longitudinal axis thereof. Thus, in
drilling a well along a specific formation stratum (which may be
horizontal or inclined), the present invention will act to indicate
to the driller whether the drill bit is approaching a stratum
boundary, and in which direction that boundary lies. With this
information, the driller can take appropriate action to change the
direction of the well bore as necessary to achieve the preselected
and desired path.
Inventors: |
Grosso; Donald S. (West
Hartford, CT), Duckworth; Allen (Middlefield, CT) |
Assignee: |
Teleco Oilfield Services Inc.
(Meriden, CT)
|
Family
ID: |
25409396 |
Appl.
No.: |
06/898,409 |
Filed: |
August 20, 1986 |
Current U.S.
Class: |
340/853.4;
175/45; 324/373; 340/856.3; 324/369; 340/856.1 |
Current CPC
Class: |
G01V
3/20 (20130101) |
Current International
Class: |
G01V
3/18 (20060101); G01V 3/20 (20060101); G01V
003/20 (); E21B 047/02 () |
Field of
Search: |
;324/323,356,369,373
;175/40,45 ;166/250,251,254,255 |
References Cited
[Referenced By]
U.S. Patent Documents
Primary Examiner: Eisenzopf; Reinhard J.
Assistant Examiner: Snow; Walter E.
Attorney, Agent or Firm: Fishman, Dionne & Cantor
Claims
What is claimed is:
1. A method of determining the existence and location of
substantially horizontal strata boundary in a geological strata
while drilling boreholes therein with a drill string segment having
a longitudinal axis, said drill string segment including a focussed
formation resistivity sensor, comprising the steps of:
generating a current field which is asymmetric with respect to the
longitudinal axis of the drill string segment, the asymmetric
current field being generated by the focussed formation resistivity
sensor;
sensing the presence of a substantially horizontal strata boundary
by analyzing measurements of formation resistance determined by
said asymmetric current field generated by said focussed formation
resistivity sensor;
sensing the location of said substantially horizontal strata
boundary relative to said drill string by means of analyzing said
measurements taken from said focussed formation resistivity sensor;
and
maintaining said drill string within said substantially horizontal
strata during the drilling of a borehole in response to the sensed
presence and location of the substantially horizontal strata
boundary.
2. The method of claim 1 wherein said focussed formation
resistivity sensor is manufactured from the steps of:
forming a drill string segment having a longitudinal axis;
surrounding a layer of electrical insulation about at least a
portion of said segment along the length thereof; and
positioning at least two electrode means for measuring the apparent
resistivity of a borehole, said electrode means being positioned at
predetermined locations along a single side of said segment in a
line parallel to the longitudinal axis of said segment, said
electrode means being disposed in said layer of insulation and
occupying a small area on the surface of said segment wherein said
electrode means generate a current field which is asymmetric with
respect to said longitudinal axis of said drill string segment.
3. The method of claim 2 including:
disposing gamma ray sensor means in said drill string segment;
and
positioning gamma ray shield means at a predetermined location in
said layer of insulation, said shield means being in alignment with
said gamma ray sensor means.
4. The method of claim 3 including:
forming said shield means of lead or tungsten.
5. The method of claim 2 including:
providing a plurality of independent pairs of said electrode means,
each pair being evenly spaced from the other pairs about the
circumference of said drill string segment.
6. The method of claim 5 including:
disposing gamma ray sensor means in said drill string segment;
and
positioning gamma ray shield means at a predetermined location in
said layer of insulation, said shield means being in alignment with
said gamma ray sensor means.
7. The method of claim 6 including:
forming said shield means of lead or tungsten.
8. The method of claim 5 including:
connecting means to said electrode means for determining the
direction of a nearby boundary in a geological strata while said
drill string is rotating about its longitudinal axis
9. The method of claim 1 including:
connecting means to said electrode means for determining the
direction of a nearby boundary in a geological strata while said
drill string is rotating about its longitudinal axis.
10. A method of determining the existence and location of
substantially horizontal coal seam boundary in a geological strata
while drilling boreholes therein with a drill string segment having
a longitudinal axis, said drill string segment including a focussed
formation resistivity sensor, comprising the steps of:
generating a current field which is asymmetric with respect to the
longitudinal axis of the drill string segment, the asymmetric
current field being generated by the focussed formation resistivity
sensor;
sensing the presence of a substantially horizontal coal seam
boundary by analyzing measurements of formation resistance
determined by said asymmetric current field generated by said
focussed formation resistivity sensor;
sensing the location of said substantially horizontal coal seam
boundary relative to said drill string by means of analyzing said
measurements taken from said focussed formation resistivity sensor;
and
maintaining said dril string within said substantially horizontal
coal seam during the drilling of a borehole in response to the
sensed presence and location of the substantially horizontal coal
seam boundary.
11. The method of claim 10 wherein said focussed formation
resistivity sensor is manufactured from the steps of:
forming a drill string segment having a longitudinal axis;
surrounding a layer of electrical insulation about at least a
portion of said segment along the length thereof; and
positioning at least two electrode means for measuring the apparent
resistivity of a borehole, said electrode means being positioned at
predetermined locations along a single side of said segment in a
line parallel to the longitudinal axis of said segment, said
electrode means being disposed in said layer of insulation and
occupying a small area on the surface of said segment wherein said
electrode means generate a current field which is asymmetric with
respect to said longitudinal axis of said drill string segment.
12. The method of claim 11 icluding:
disposing gamma ray sensor means in said drill string segment;
and
positioning gamma ray shield means at a predetermined location in
said layer of insulation, said shield means being in alignment with
said gamma ray sensor means.
13. The method of claim 12 including:
forming said shield means of lead or tungsten.
14. The method of claim 11 including:
providing a plurality of independent pairs of said electrode means,
each pair being evenly spaced from the other pairs about the
circumference of said drill string segment.
15. The method of claim 14 including:
disposing gamma ray sensor means in said drill string segment;
and
positioning gamma ray shield means at a predetermined location in
said layer of insulation, said shield means being in alignment with
said gamma ray sensor means.
16. The method of claim 15 including:
forming said shield means of lead or tungsten.
17. The method of claim 14 including:
connecting means to said electrode means for determining the
direction of a nearby boundary in a geological strata while said
drill string is rotating about its longitudinal axis.
18. The method of claim 11 including:
connecting means to said electrode means for determining the
direction of a nearby boundary in a geological strata while said
drill string is rotating about its longitudinal axis.
Description
BACKGROUND OF THE INVENTION
This invention relates to a device for use in the drilling of
directional (i.e., non-vertical) boreholes or wells. More
particularly, this invention relates to a device for use in
navigating and guiding the well being drilled along a particular
stratum by detecting the approach to a boundary of the stratum and
the direction of said approach. The device of the present invention
detects this information which is then used to change the direction
of the well away from the boundary of the stratum and back towards
the preselected and desired path.
In many applications, it is necessary and desirable to drill a
section of a well along a particular stratum. This stratum may be
horizontal or inclined and therefore the well which is drilled
therethrough must be guided so as not to deviate into adjacent
strata. Accordingly, the directional driller must be provided with
information which indicates when the drill is approaching a
boundary between two adjacent strata, and from which direction the
approach is occurring. One particularly well known application of
this type of directional drilling is found in coal mining wherein
boreholes are desired within a seam of coal which is typically in a
horizontal or inclined stratum and is sandwiched between layers of
shale or other materials. In this application, it is desired to
keep the well or bore hole within the coal seam and therefore the
directional driller must know when the well is approaching a
shale/coal interface so that the well can be directed back into the
coal seam.
One known prior art method of directional drilling within a
particular stratum utilizes a sensing and detector device which is
based on a gamma radiation sensor. Significantly, unlike other
known gamma radiation sensors, the gamma radiation sensor used in
directional drilling within a particular stratum utilizes a
tungsten or lead shield covering a portion of its circumference. As
a result, gamma radiation from one direction is sensed at full
strength while radiation from the opposite direction is minimized
by the shield. This device makes use of the fact that adjacent
strata generate different amounts of radiation. Thus, by rotating
the gamma radiation sensor about its axis, a nearby boundary
between strata may be detected. The direction of this boundary is
determined by rotating the sensor and measuring the angle at which
the radiation received is maximum or minimum.
While suitable for its intended purposes, there is a perceived need
in the art for additional means for measuring the boundary layer
between adjacent strata during such directional drilling. This
additional sensing means may be used both separately or in
conjunction with exisiting gamma radiation sensor devices. An
additional independent sensing means would provide an important
backup system to gamma radiation systems and limit the possibility
for error.
SUMMARY OF THE INVENTION
The above-discussed and other problems and deficiencies of the
prior art are overcome or alleviated by the sensing and detecting
device for use in directional drilling of the present invention. In
accordance with the present invention, a device for detecting and
sensing the boundaries between differing rock strata is provided.
This device utilizes electrical conductivity or resistivity to
detect and measure the boundary layer. The device of the present
invention may be used with a "measurement-while-drilling" (MWD)
tool, using either wire-line or wireless data communication with a
surface receiver. Moreover, the resistivity or conductivity
detection device of the present invention may also be used alone or
in conjunction with prior art gamma radiation sensors. The present
invention utilizes at least a pair of electrodes which occupy a
small area and are located on one side of the tool aligned with the
longitudinal axis thereof.
Thus, in drilling a well along a specific formation stratum (which
may be horizontal or inclined), the present invention will act to
indicate to the driller whether the drill bit is approaching a
stratum boundary, and in which direction that boundary lies. With
this information, the driller can take appropriate action to change
the direction of the well bore as necessary to achieve the
preselected and desired path.
The above discussed and other features and advantages of the
present invention will be appreciated and understood to those
skilled in the art from the following detailed description and
drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
Referring now to the drawings, wherein like elements are numbered
alike in the several FIGURES:
FIG. 1A is a front elevation view, partly in cross section, of a
portion of a formation resistivity tool in accordance with the
prior art;
FIG. 1B is a cross-sectional elevation view along the line 1B--1B
of FIG. 1A;
FIG. 2A is a front elevation view, partly in cross-section, of a
focussed formation resistivity tool in accordance with the present
invention;
FIG. 2B is a cross-sectional elevation view along the line 2B--2B
of FIG. 2A;
FIG. 3A is a schematic view with lines indicating the flow of
current in a formation resistivity tool in accordance with the
prior art;
FIG. 3B is a plan view along the line 3B--3B of FIG. 3A;
FIG. 4A is a schematic view showing the lines of current flow in a
focussed formation resistivity tool in accordance with the present
invention;
FIG. 4B is a plan view along the line 4B--4B of FIG. 4A;
FIG. 5A is a graphical representation of a rotary drilling log
plotting depth versus resistivity;
FIG. 5B is a graph plotting resistivity versus tool face angle;
FIG. 6 is a front elevation view, partly in cross-section, of a
focussed gamma and formation resistivity tool in accordance with
the present invention and in conjunction with a mud pulse telemetry
system;
FIG. 7 is a cross-sectional elevation view along the line 7--7 of
FIG. 6;
FIG. 8 is a front elevation view of an alternative embodiment of
the present invention; and
FIG. 9 is a cross-sectional elevation view along the line 9--9 of
FIG. 8.
DESCRIPTION OF THE PREFERRED EMBODIMENT
Many successful and well known instruments have been used for
measuring the electrical conductivity (or resistivity) of earth
formations in boreholes. Several of these prior art formation
resistivity devices have been adapted for use with logging or
measurement-while-drilling (MWD) systems. It will be appreciated
that the present invention is similar to prior art formation
resistivity instruments, but has been improved for the specific
purpose of navigating a drill bore along a specified stratum,
particularly a stratum which is horizontal or inclined (rather than
vertical).
Referring first to FIGS. 1A and 1B, a portion of a formation
resistivity tool in accordance with the prior art is shown
generally at 10. Thus, FIG. 1 shows a MWD version of the "short
normal" resistivity sensor as is used in conventional
logging-while-drilling operations. Formation resistivity tool 10 is
shown disposed in a borehole 12, borehole 12 extending across
several formation boundaries identified at 14, 15, and 16. The tool
of FIG. 1 is optimized for drilling across the formation boundaries
14, 15, 16 and 17. The schematic cross-sectional view of formation
resistivity tool 10 includes a plurality of symmetrical electrode
rings 18A and 18B which are mounted in an insulated jacket 20.
Jacket 20 is mounted on a drill collar 22 which surrounds a passage
24 and an instrument housing 26. It will be appreciated that
drilling mud flows through passage 24 to get to the drill bit at
the bottom of the borehole from the surface.
In FIGS. 2A and 2B, a "focussed" formation resistivity sensor
device in accordance with the present invention is shown generally
at 28. Unidirectional or focussed formation resistivity tool 28 is
similar to the prior art tool 10 shown in FIGS. 1A and 1B with the
exception that rather than electrode 18 having a ring shape and
surrounding the entire drill collar as in the prior art, the
electrodes 30A and 30B of the present invention occupy a small area
and are only located on one side of the tool. It will be
appreciated that the remaining structural features of tool 28 and
FIGS. 2A and 2B are the same or similar to the tool 10 in FIGS. 1A
and 1B and so are indicated using the same numerals with the
addition of a prime. Also, unlike tool 10 which is shown disposed
in a borehole vertically extending across formation boundaries,
tool 28 of FIG. 2A is shown in a single stratum layer between
boundaries of adjacent strata. By way of example only and for
purposes of ease of discussion, the focussed formation resistivity
tool 28 in FIG. 2A is located in a borehole 12' which is in a seam
of coal 32, coal 32 being sandwiched between two layers of shale 34
and 36. Thus, a boundary layer 38 is formed between shale layer 34
and coal seam 32 while a boundary layer 40 is formed between shale
layer 36 and coal seam 32.
Referring jointly to FIGS. 1-4, in both the prior art formation
resistivity tool 10 and the focussed formation resistivity tool of
the present invention, the method of obtaining the measurement of
formation resistivity is the same. That is, tool electrodes are
located on the outside of the tool housing which are insulated from
the housing and from each other. The first electrode transmits an
electric current (either direct current or a low frequency
alternating current) into the borehole fluid, which must be
electrically conductive, and into the surrounding formations. This
current returns to the metallic, non-insulated housing of the tool.
The second electrode measures the electric potential surrounding
it. This potential is a measure of the current flowing in the
formation, which in turn, is determined by the resistivity of the
formation.
In the conventional logging-while-drilling tool shown in FIGS. 1A
and 1B, the electrodes 18A and 18B are in the form of circular
bands which surround the circumference of the tool 10. As shown in
FIGS. 3A and 3B, this electrode configuration generates a pattern
of current flow which is symmetrical about the axis (identified at
0) of the tool, and the response of the sensor is axisymmetric.
Significantly, this axisymmetric response will be true whether the
tool is rotating about its axis or whether the tool is stationary
while the measurement is being made.
In contrast, in the present invention, the two electrodes 30A and
30B are in the form of small surfaces centered on a line parallel
to the axis of the tool rather than as in two concentric or
circular bands. As a result, the electrode configuration generates
a distinctly asymmetrical current field as shown in FIGS. 4A and
4B. Similarly, the response of the sensor is also asymmetric. It
will be appreciated that the details of the methods of current
generation and potential measurement will be the same for both the
prior art tool and the present invention. Moreover, the manner in
which such current generation is made and the potential measurement
taken is well known to those skilled in the art (for example, see
U.S. Pat. No. 4,570,123 which is assigned to the assignee hereof
and incorporated herein by reference thereto) and so no further
discussion is necessary.
The unidirectional or asymetric response of the focussed
resistivity sensor of the present invention is well suited for
those drilling applications wherein assistance is needed in the
navigation of the borehole along a particular stratum. A method of
operating the tool of the present invention involves rotating the
tool about its axis while drilling (as indicated by the arrows 39
and 41 in FIG. 2A). It will be appreciated that
rotating-while-drilling is a normal practice for rotary drilling
operations. While such rotating-while-drilling is being done,
measurements of formation resistance are accumulated over a period
of greater than one revolution of the tool. A graphical "log" is
produced while drilling. If the borehole is passing through a
homogenous stratum of reasonable depth, and the measurement is
integrated over more than one revolution, the readings will be
constant. This condition is shown in FIG. 5A wherein a rotary
drilling log plotting depth versus resistivity is shown. It will be
appreciated that constant readings indicating a homogenous stratum
is identified in the area marked A. However, if the borehole
approaches a boundary with a different formation, the readings will
begin to deviate from the previous constant value. This deviation
is shown as an increase in reading as identified in FIG. 5A. Such a
deviation will indicate to the drilling operator that the borehole
is approaching a boundary layer (i.e., 38 or 40 in FIG. 2A), of the
stratum.
While the driller now knows that a boundary layer between strata is
being approached, information is still necessary as to the
direction in which that boundary layer lies relative to the tool in
the borehole (i.e., the direction of approach). To measure this
information, tool 28 is held stationary in the borehole while a
resistivity reading is made. At that time, a directional survey is
conducted by a sensor attached to the resistivity sensor. These
measurements are then transmitted to the surface by the MWD tool
and displayed, preferably graphically, by the surface receiver.
This information establishes the direction in which the sensitive
axis of the resistivity sensor is inclined. The tool is then moved
to a new position by rotating it about its axis (this can easily be
done from the surface by rotating the drill string as a whole). A
new set of readings is made at this new angle, and the process is
repeated several times. These measurements are then analyzed to
determine the direction in which the maximum and minimum values of
resistivity lie (as the maximum and minimum values of resistivity
indicate the direction of the stratum boundary which is being
sought). A convenient method of interpreting this data is by means
of a graph. Thus, referrring to FIG. 5B, the resistivity
measurements are plotted against the corresponding values of sensor
direction "tool face angle". Because the form of the curve (e.g. a
sine wave) is known from previous calibrations of the instrument,
only a few data points (i.e. 90.degree., 180.degree., 270.degree.,
and 360.degree.) are required to establish the complete curve and
hence the desired angle defining the direction of the stratum
boundary. The curve of the graph may be established by manual
means, or most conveniently by a computer curve fitting program. In
the example shown in FIG. 5B, the boundary of a more resistive
stratum is located at a tool face angle of 180.degree., i.e.
upward. Thus, in order to avoid drilling through this boundary
layer, the drilling procedure must be changed to cause the drill to
take a more downward direction. If the response of the resistivity
sensor to the formations concerned are well known such as in
coal/shale strata, the difference between the highest and lowest
resistivity values may be used to calculate how far away the
boundary is from the survey point in the well bore. The severity of
the angle change required to correct the well path can then be
assessed.
The focussed formation resistivity tool of the present invention
can be used alone with an MWD tool or may also be used in
conjunction with prior art "focussed" gamma radiation detectors.
Referring to FIG. 6, apparatus for MWD resistivity logging is
schematically illustrated. The apparatus is a "focussed"
resistivity gamma directional (RGD) MWD measurement tool similar to
an MWD tool presently in commercial use by Teleco Oilfield
Services, Inc. (assignee of the present invention). The apparatus
shown in FIG. 6 is a drill collar having an upper steel section 46
and a lower non-magnetic sub 48, sections 46 and 48 being threaded
together, with appropriate electrical interconnections.
As mentioned, the "focussed" RGD tool of FIG. 6 consists of two
sections 46 and 48. The upper portion of the tool houses Teleco's
standard directional sensor 50, turbine/alternator 52, transmitter
54 and microprocessor and electronics 56. The downhole
microprocessor 56 processes the logging data and controls the
transmission sequence continuous formation gamma and resistivity
data while rotating, directional data while not rotating. The
transmitter 54 is a mud pulse transmitter as shown and described in
U.S Pat. Nos. 3,982,431, 4,013,945; and 4,021,774, assigned to the
assignee hereof and incorporated herein in their entirety.
An auxiliary sensor module is mounted in sub 48. Located in the
center of this unit is the gamma ray detector (scintillation
crystal) 58 and an electronics package 60. A gamma radiation shield
62 (usually comprised of tungsten or lead) covers a portion of the
circumference of sub 48. In this way, gamma radiation from one
direction is sensed at full strength while radiation from the other
direction is minimized by the shield 62. The collar surface has an
insulated region 64 with two uni-directional or focussed electrodes
66 and 68 which are used to make the resistivity measurements.
Electrodes 66 and 60 are positioned along a single side of sub 48
and occupy only a small area or surface thereof. Electrodes 66 and
60 are centered on a line parallel to the axis of the tool and may
be any desired shape including, but not limited to diamonds,
circles, rectangles, triangles, etc. Electrical connection of the
resistivity electrodes to electronics 60 may be accomplished in any
standard manner known in the art. The geometry of the tool places
the measured points of the "focussed" gamma ray and resistivity
sensors at the same depth. This facilitates analysis and allows
both measurements to be made in a zone of interest at the same
time.
The "focussed" gamma ray formation resistivity tool of FIG. 6 has
many features and advantages over prior art tools used in
applications wherein direction drilling within a particular stratum
and without deviating into adjacent strata is necessary. As
mentioned hereinabove, prior art tools have heretofore utilized
only focussed gamma ray detectors. Such devices make use of the
fact that adjacent strata generate different amounts of radiation.
Thus, by rotating this sensor about its axis, a nearby bed boundary
may be sensed. The direction of the boundary is determined by
rotating the sensor and measuring the angle at which the radiation
received is maximum or minimum. While suitable for its intended
purposes, the "focussed" gamma radiation detectors of FIG. 6 may
fail through breakdown or may provide inaccurate data due to the
presence of other radioactive materials.
The "focussed" formation resistivity sensor of the present
invention provides an alternative method of detecting the presence
and direction of boundaries between adjacent strata which operates
at least as effectively as "focussed" gamma radiation sensors.
Moreover, in a preferred embodiment, a combined MWD sensor tool
such as shown in FIG. 6 which incorporates both focussed gamma and
focussed resistivity sensors will provide alternative and/or
complementary systems in the same tool. Thus, two independent
readings may be checked against each other with the result being a
higher confidence level, less down time and better drilling
results.
As discussed with regard to FIGS. 5A and 5B, the tool must be
rotated about its axis in order to determine the direction of the
stratum boundary being approached. However, in certain drilling
operations, particularly those operations employing mud motor
assemblies, rotation of the drill string is undesired or not
possible.
Referring now to FIGS. 8 and 9, an alternative embodiment of the
present invention is shown at 70 wherein a focussed resisitivity
tool uses a plurality of independent small electrode pairs around
the circumference of the tool, with each electrode pair located on
one side of the tool and aligned with the longitudinal axis
thereof. Focussed formation resistivity tool 70 is similar to the
tools shown in FIGS. 2 and 6 with the difference being that the
additional electrode pairs are positioned at evenly spaced
intervals. In the example shown in FIGS. 8 and 9, six independent
pairs of electrodes 72, 74, 76, 78 and 82 are positioned at 60
degree intervals about the circumference of the tool. Each
independent electrode pair is then actuated separately and the
formation continuum is evaluated in relation to the tool heading
without having to rotate the tool. Each independent pair of
electrodes is electrically energized independently of the other
pairs upon command by the tool electronics. Thus, the embodiment of
FIGS. 8 and 9 has the advantage of being able to detect the
direction in which the stratum boundary lies without rotation of
the tool or drill string. This permits the present invention to be
used in conjunction with a mud rotor assembly while concurrently
examining the formation continuum.
A further enhancement when drilling in the rotary drilling mode,
using either the plurality of electrode pairs or the single
electrode pair embodiments, is to make the resistivity and
directional measurements continuously while rotating and reckon
formation discontinuities to the tool heading with a microprocessor
located downhole within the tool and transmit only the vector
heading of the formation discontinuity. This is accomplished in the
plurality electrode pair configuration, (FIGS. 8 and 9) by
measuring all electrode pairs at the same time and in the single
electrode pair (FIG. 6) by measuring multiple points during one
revolution of the drillstring as triggered by an azimuth sensor.
The microprocessor then computes the vector using a curve fitting
routine and a maximum/minimum routine (the point where the first
derivative goes through zero).
While preferred embodiments have been shown and described, various
modifications and substitutions may be made thereto without
departing from the spirit and scope of the invention. Accordingly,
it is to be understood that the present invention has been
described by way of illustrations and not limitation.
* * * * *